YES

Problem:
 f(f(X)) -> f(g(f(g(f(X)))))
 f(g(f(X))) -> f(g(X))

Proof:
 DP Processor:
  DPs:
   f#(f(X)) -> f#(g(f(X)))
   f#(f(X)) -> f#(g(f(g(f(X)))))
   f#(g(f(X))) -> f#(g(X))
  TRS:
   f(f(X)) -> f(g(f(g(f(X)))))
   f(g(f(X))) -> f(g(X))
  EDG Processor:
   DPs:
    f#(f(X)) -> f#(g(f(X)))
    f#(f(X)) -> f#(g(f(g(f(X)))))
    f#(g(f(X))) -> f#(g(X))
   TRS:
    f(f(X)) -> f(g(f(g(f(X)))))
    f(g(f(X))) -> f(g(X))
   graph:
    f#(g(f(X))) -> f#(g(X)) -> f#(g(f(X))) -> f#(g(X))
    f#(f(X)) -> f#(g(f(g(f(X))))) -> f#(g(f(X))) -> f#(g(X))
    f#(f(X)) -> f#(g(f(X))) -> f#(g(f(X))) -> f#(g(X))
   SCC Processor:
    #sccs: 1
    #rules: 1
    #arcs: 3/9
    DPs:
     f#(g(f(X))) -> f#(g(X))
    TRS:
     f(f(X)) -> f(g(f(g(f(X)))))
     f(g(f(X))) -> f(g(X))
    Bounds Processor:
     bound: 1
     enrichment: match
     automaton:
      final states: {9,4,1}
      transitions:
       f30() -> 2*
       f{#,0}(3) -> 1*
       g0(5) -> 6*
       g0(7) -> 8*
       g0(2) -> 3*
       f0(2) -> 5*
       f0(6) -> 7*
       f0(8) -> 4*
       f0(3) -> 9*
       f1(17) -> 18*
       f1(14) -> 15*
       g1(30) -> 31*
       g1(34) -> 35*
       g1(24) -> 25*
       g1(16) -> 17*
       g1(28) -> 29*
       g1(13) -> 14*
       2 -> 16*
       3 -> 34*
       4 -> 5*
       6 -> 13*
       8 -> 30*
       9 -> 5*
       14 -> 28*
       15 -> 4*
       17 -> 24*
       18 -> 7*
       25 -> 14*
       29 -> 17*
       31 -> 17*
       35 -> 17*
     problem:
      DPs:
       
      TRS:
       f(f(X)) -> f(g(f(g(f(X)))))
       f(g(f(X))) -> f(g(X))
     Qed